##
**Phase operators, temporally stable phase states, mutually unbiased bases and exactly solvable quantum systems.**
*(English)*
Zbl 1186.81052

Summary: We introduce a one-parameter-generalized oscillator algebra \(\mathcal A_k\) (that covers the case of the harmonic oscillator algebra) and discuss its finite- and infinite-dimensional representations according to the sign of the parameter \(\kappa \). We define an (Hamiltonian) operator associated with \(\mathcal A_k\) and examine the degeneracies of its spectrum. For the finite (when \(\kappa < 0\)) and the infinite (when \(\kappa \geq 0\)) representations of \(\mathcal A_k\), we construct the associated phase operators and build temporally stable phase states as eigenstates of the phase operators. To overcome the difficulties related to the phase operator in the infinite-dimensional case and to avoid the degeneracy problem for the finite-dimensional case, we introduce a truncation procedure which generalizes the one used by Pegg and Barnett for the harmonic oscillator. This yields a truncated-generalized oscillator algebra \(\mathcal A_{\kappa ,s}\), where \(s\) denotes the truncation order. We construct two types of temporally stable states for \( \mathcal A_{\kappa ,s}\) (as eigenstates of a phase operator and as eigenstates of a polynomial in the generators of \(\mathcal A_{\kappa ,s})\). Two applications are considered in this paper. The first concerns physical realizations of \(\mathcal A_{\kappa}\) and \(\mathcal A_{\kappa ,s}\) in the context of one-dimensional quantum systems with finite (Morse system) or infinite (Pöschl-Teller system) discrete spectra. The second deals with mutually unbiased bases used in quantum information.

### MSC:

81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |

33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |

11C08 | Polynomials in number theory |

81R15 | Operator algebra methods applied to problems in quantum theory |

81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |

81R10 | Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations |